Fuzzy Bag Models

We show how hadronic bag models can be generalized to implement effects of a smooth and extended boundary. Our approach is based on fuzzy set theory and can be straightforwardly applied to any type of bag model. We illustrate the underlying ideas by calculating static nucleon properties in a fuzzy chiral bag model. Typeset using REVTEX HD-TVP 97-12 Supported by habilitation grant Fo 156/2-1 from Deutsche Forschungsgemeinschaft. 1 Physical concepts and models are often based on idealizations. Usually, those arise either from insufficient knowledge of the underlying physics, or they are intended to make the theoretical description more transparent and more amenable to quantitative analysis. Bag models [1,2], which occupy a prominent place among hadron models and are widely used in areas ranging from hard scattering processes to dense nuclear matter, furnish a typical example for such idealizations. They impose the confinement of relativistic quarks inside hadrons, in a region of modified vacuum, by static boundary conditions at a bag radius R. Whereas the real vacuum is expected to return to its normal phase outside of the hadron gradually, however, this simple prescription leads to an infinitely thin bag boundary and thus to an abrupt transition between the two phases. Of course, such a rough and energetically unfavorable approximation misses some relevant features of the physics of hadrons. Especially observables with an exceptional sensitivity to the characteristics of the boundary, such as for example some properties of excited and deformed hadrons or diffractive scattering cross sections (in particular at low energies), therefore require a more realistic description of the hadronic boundary. Previous attempts to go beyond the sharp bag-boundary approximation, however, were technically quite involved and limited to a specific model [3]. In the present letter we consider a novel implementation of extended boundaries, which is easy to apply to even the most complex bag models (including those with quantized surfaces). This approach can be rigorously formulated in terms of fuzzy set theory [4,5], in which ordinary sets are generalized by assigning partial memberships to their elements. By now, fuzzy sets have proven remarkably useful in quite diverse areas of model building, and it seems worthwhile and timely to explore their potential in physics. The application to the transition between the inside and outside regions of bag models suggests itself naturally since fuzzy sets were specifically designed to implement smooth transitions between unrealistically distinct domains in simplified models. It is quite straightforward to see how such fuzzy boundaries arise. To start with, one considers the sharp surface of the standard bag model at a given radius as the sole element 2 of an ordinary set. By letting this set become fuzzy, an extended boundary – containing conventional bag surfaces of varying radii and weights as elements – emerges. In analogy with the boundary conditions of standard bag models, the underlying fuzzy set (the weight function) is prescribed according to general physical requirements. Possibly, it could be determined dynamically in a future, more advanced version of the model. As just indicated, the central idea of our approach is to promote the bag radius from a real number R to a fuzzy set ρ. In general, fuzzy sets [4] consist of an ordinary reference set X and a real-valued membership function μ : X → [0, 1] x 7→ μ(x) , (1) which specifies the degree to which an element x ∈ X belongs to μ. (Following common practice, we use the same symbol for both the fuzzy set and its membership function.) By definition, μ is an element of the fuzzy power set F(X ) over X . Taken as the truth value of a statement x, μ(x) defines a generalization of Boolean logic (called L1 [6]) in which the strict true-false alternative for x is relaxed. Accordingly, the fuzzy bag radius is represented by a membership function ρ(R), which specifies the degree to which a sphere with radius R belongs to the extended bag boundary. Therefore, its reference set R ⊆ [0,∞] minimally contains the radii in the surface region. We denote the center (in radial direction) of the boundary by R0 and its width by ∆. Some of the potential of this description of the boundary originates from the fact [5] that membership degrees in fuzzy sets are generally not additive (in contrast, for example, to probabilities). This implies, e.g., that bag surfaces at different R (i.e. their fuzzy weights) do not have 1This can be seen directly from the membership degree of subsets R1 ∈ R, which is given by ρ(R1) = sup{ρ(R)|R ∈ R1} [4]. For the same reason, ∫ X μ(x) dx 6= 1 in general. The probability P (R1) = ∫